One-way or Two-way Factor Model for Matrix Sequences?

TL;DR

This paper develops new statistical tests to determine whether matrix-valued data has a one-way or two-way factor structure, using eigen-gap analysis and randomization techniques, applicable without restrictive assumptions.

Contribution

It introduces a family of eigen-gap based, non-thresholding tests for identifying one-way or two-way factor models in matrix sequences, with a de-randomized decision rule and improved small-sample performance.

Findings

Tests perform well in large samples.

Projection-based method outperforms existing methods in small samples.

Proposed procedures accurately estimate the number of factors.

Abstract

This paper investigates the issue of determining the dimensions of row and column factor spaces in matrix-valued data. Exploiting the eigen-gap in the spectrum of sample second moment matrices of the data, we propose a family of randomised tests to check whether a one-way or two-way factor structure exists or not. Our tests do not require any arbitrary thresholding on the eigenvalues, and can be applied with no restrictions on the relative rate of divergence of the cross-sections to the sample sizes as they pass to infinity. Although tests are based on a randomization which does not vanish asymptotically, we propose a de-randomized, strong (based on the Law of the Iterated Logarithm) decision rule to choose in favor or against the presence of common factors. We use the proposed tests and decision rule in two ways. We further cast our individual tests in a sequential procedure whose output is an estimate of the number of common factors. Our tests are built on two variants of the sample second moment matrix of the data: one based on a row (or column) flattened version of the matrix-valued sequence, and one based on a projection-based method. Our simulations show that both procedures work well in large samples and, in small samples, the one based on the projection method delivers a superior performance compared to existing methods in virtually all cases considered.